1. Fluid Mechanics, KU, 2011 Chap. 8: One-Dimensional Flows Incompressible Newtonian fluids considered here EOC + Navier-Stokes eq. (EOM with Newtonian CE) Simplification of governing equations by discarding procedure (Physically reasonable assumptions are employed) Procedure for obtaining solutions: - Understand velocity field depending on independent variables - Check EOC or utilize it - Solve Navier-Stokes eq. 1. Introduction 2. Plane Poiseuille flow Problem description Problem description Steady state / incompressible Newtonian fluids / large aspect ratio (H / W  0) / no entrance & exit effect (H/L << 1) / laminar flow fully developed !

2. Fluid Mechanics, KU, 2011 Direct solution Direct solution Fully developed: Infinite in z-direction:

3. Fluid Mechanics, KU, 2011 steady state P ~ indep. of y and z coord.  P = P(x)

4. Fluid Mechanics, KU, 2011 Integration for v x over y Two unknowns  two B.C.s needed Symmetry boundary condition Symmetry boundary condition Agreement with experiments up to for W/H > 10 Solution logic Solution logic Positive

5. Fluid Mechanics, KU, 2011 3. Plane Couette flow

6. Fluid Mechanics, KU, 2011 P = P(x) But, P=const. from Eng. Bernoulli eq. Constant !

7. Fluid Mechanics, KU, 2011 4. Poiseuille flow Problem description Problem description

8. Fluid Mechanics, KU, 2011 P ~ indep. of r and  coord.  P = P(z) Integration over r

9. Fluid Mechanics, KU, 2011

10. 5. Wire coating Find R c from velocity profile

11. Fluid Mechanics, KU, 2011 Indep. of viscosity

12. Fluid Mechanics, KU, 2011 6. Torsional flow No wall effect L >> R, no end effect at bottom & free liquid interface No axial and radial motion No imposed pressure gradient, Velocity field EOMs

13. Fluid Mechanics, KU, 2011 Torque Pressure Min. pres at r=R (cylinder surface) Negative: torque exerted on the cylinder by the fluid

14. Fluid Mechanics, KU, 2011 7. Rectilinear Flow & Hydraulic Diameter Velocity field EOMs

15. Fluid Mechanics, KU, 2011 Dimensionless velocity : u Dimensionless distance : ξ Hydraulic diameter : D H

16. Fluid Mechanics, KU, 2011 8. Tube Flow of a Power-Law Fluid EOMs Power-Law Fluid

17. Fluid Mechanics, KU, 2011